Conflict-Based Search (CBS)

Updated 30 November 2025

Conflict-Based Search (CBS) is a two-level, decoupled optimal framework for multi-agent path finding that separates conflict resolution from individual agent planning.
It employs a high-level constraint tree to resolve agent conflicts and a low-level planner to generate individual paths under dynamic constraints.
The framework adapts to continuous, heterogeneous, and kinodynamic systems, offering practical scalability with rigorous guarantees of completeness and optimality.

Conflict-Based Search (CBS) is a two-level, decoupled optimal algorithmic framework for multi-agent path finding (MAPF) and, with suitable adaptations, for broader multi-agent motion planning (MAMP) in continuous or heterogeneous spaces. CBS is notable for its scalability, completeness, and ability to decouple high-dimensional joint planning into independent low-level searches punctuated by conflict identification and constraint propagation. The CBS framework admits a broad taxonomy of extensions, enhancements, and theoretical analyses, encompassing probabilistic settings, kinodynamic constraints, continuous time and space, multi-objective criteria, explainability, meta-agent formation, and integration with heterogeneous or learned solvers.

1. Core Principles and Formalization

CBS operates on two levels interconnected by a constraint tree (CT) (Solis et al., 2019, Tolpin, 2014):

High-Level (CT Search): The high-level search tree models the combinatorics of agent–agent conflicts. Each CT node consists of a set of constraints $C$ , a set of single-agent paths $P=\{\rho_1, ..., \rho_n\}$ (one per agent), and a cost $g=\text{cost}(P)$ such as sum-of-costs (SOC) or makespan (MS). Constraints typically take the form ⟨i, v, t⟩ (agent $i$ forbidden at vertex $v$ at time $t$ ) for vertex conflicts, or an edge constraint ⟨i, (u→v), t⟩ (agent $i$ forbidden to traverse $u\rightarrow v$ at time $t$ ) for edge swap conflicts (Solis et al., 2019, Tolpin, 2014).
Low-Level (Single-Agent Planner): Each agent is planned independently with its own shortest path or optimal trajectory under its set of constraints. For discrete MAPF, this is typically an A* or Dijkstra search on time-expanded graphs; for continuous and heterogeneous cases, a per-agent roadmap or sampling-based planner is used, with constraints interpreted as dynamic obstacles or forbidden configurations (Solis et al., 2019, Kottinger et al., 2022, Veerapaneni et al., 1 Oct 2025).

A standard high-level procedure iteratively expands the lowest-cost CT node, inspects for the first agent–agent conflict, and branches by augmenting the constraints for each involved agent in turn, replanning only that agent. The process continues until a conflict-free node is found, at which point its associated joint plan is returned as optimal under the team cost metric (Solis et al., 2019).

2. Constraint Handling and Extensions to Heterogeneous, Continuous, and Kinodynamic Systems

CBS generalizes naturally to handle:

Continuous State-Spaces (MAMP, High-DOF): Each agent operates on a distinct PRM or roadmap in its own configuration space $\mathcal{C}_i$ . Time synchronization is imposed via a global discretization $\Delta t$ , yielding compatible timelines for heterogeneous agents. Constraints extend to ⟨i, q, t⟩, forbidding agent $i$ from occupying configuration $q$ at time $t$ ; in kinodynamic or geometric settings, these are enforced as time-stamped dynamic obstacles (Solis et al., 2019, Kottinger et al., 2022).
Kinodynamic and Sampling-Based Planners: The low-level planner can be any method capable of enforcing the imposed space–time constraints, including RRT, direct collocation, or learned policies with sufficient completeness guarantees (Kottinger et al., 2022, Veerapaneni et al., 1 Oct 2025).
NSpace–Time Constraint Protocol: Formulating CBS as a protocol, each agent is required to have a black-box Plan(i, C_i) API returning a path that respects all input space–time constraints. This allows CBS to coordinate agents with heterogeneous solvers, optimization criteria, or physical dynamics (Veerapaneni et al., 1 Oct 2025).

The constraint tree adapts to high-dimensional settings by associating constraints with the appropriate agent-dependent representation, generalizing both the semantic and physical meaning of constraints (Solis et al., 2019, Fredriksson et al., 29 Jan 2025).

3. Guarantees: Completeness, Optimality, and Representation-Optimality

Theoretical analysis establishes the following guarantees in the CBS framework (Solis et al., 2019, Kottinger et al., 2022):

Completeness: For a fixed, finite set of per-agent roadmaps or motion primitives, the high-level CBS query terminates in a finite number of steps and either returns a conflict-free set of paths or (if all possibilities are exhausted) fails.
Representation-Optimality: CBS returns a solution that is optimal with respect to the search space defined by the per-agent roadmaps. That is, no other combination of those PRMs yields a smaller team cost (Solis et al., 2019). Repeated enrichment of per-agent roadmaps (e.g., via PRM sampling) guarantees that, in the probabilistic limit, the true optimal solution is encoded.
Probabilistic Completeness: In scenarios where the roadmap is incrementally sampled (e.g., PRM-based continuous MAMP), repeated sampling and CBS querying ensures that, with probability one, a representative-optimal solution exists as the number of samples approaches infinity (Solis et al., 2019).
Soundness of Branching: At each branching step, the set of constraints is constructed so that an optimal solution must obey at least one of the children’s new constraints; the search never discards feasible solutions.
Specialization to Discrete MAPF: In the classic discrete setting, CBS recovers the standard completeness and optimality guarantees for MAPF with SOC or makespan objectives (Tolpin, 2014).

4. Algorithmic Enhancements and Variants

CBS has spawned a broad spectrum of variants and optimizations suited to specific MAPF and MAMP regimes:

Meta-Agent CBS (MA-CBS): Agents that repeatedly conflict are dynamically merged into meta-agents, such that joint low-level planning is performed for the merged entity. The merge criterion is typically a fixed or learned threshold on observed conflicts, with theoretical justification via ski-rental models (Tolpin, 2014).
Anytime and Suboptimal CBS (ECBS, CBS-Budget): Relaxing optimality via bounded suboptimal focal search, budget constraints in low-level search, or explicit estimation (Lim et al., 2022, Ivanashev et al., 2022). Implementation of dual-level or hierarchical focal search accelerates convergence to feasible solutions.
Scalability Heuristics: Practical schemes—CT size bounding with roadmap expansion, lazy collision checking, and coarsening of time-discretization—greatly improve performance while maintaining representational guarantees (Solis et al., 2019).
Conflict Selection Policies: Heuristic, lookahead, or learning-guided strategies for selecting which conflict to resolve next, reducing constraint tree size and runtime (Huang et al., 2020).
Macro-splits and Enhanced Branching: Techniques such as “head-on” splitting for inevitable conflict sequences, which batch-resolve chains of future conflicts and shrink the branching depth (Yang, 2020).
Continuous-Time and Kinodynamic Extensions: Safe Interval Path Planning (SIPP), moving obstacle constraints, and integration with generalized SIPP (GSIPP) enable direct handling of kinodynamically constrained robots, heterogeneous collision envelopes, and temporal synchronization (Kottinger et al., 2022, Andreychuk et al., 2021).

5. Complexity Analysis and Performance

Exponential Complexity: The worst-case complexity of standard CBS is exponential in the number of conflicts, scaling as $O(2^c)$ for $c$ conflicts. Formal upper bounds leverage Multi-valued Decision Diagram (MDD) representations and recurrence relations, yielding bounds dependent on the number of agents, solution cost, and graph structure (Gordon et al., 2021). Improved analysis indicates that complexity scales as $O((e n)^{kC})$ where $n$ is environment size, $k$ agent count, and $C$ the solution cost, emphasizing sensitivity to team cost rather than ambient graph size.
Practical Tractability: In sparse environments, CBS variants routinely solve problems with 10–32+ agents, high-DOF arms, and heterogeneous teams given a well-chosen per-agent roadmap representation (Solis et al., 2019, Fredriksson et al., 29 Jan 2025, Hong et al., 17 Sep 2025). The constraint-tree size can be dramatically reduced with strategic enhancements, e.g., repulsive low-level trajectory modifications in multi-manipulator planning (Hong et al., 17 Sep 2025).

6. Empirical and Application Domains

CBS and its descendants have been empirically validated in a spectrum of MAPF and multi-robot benchmarks (Solis et al., 2019, Fredriksson et al., 29 Jan 2025, Hong et al., 17 Sep 2025):

Homogeneous Agents in Large Open Fields: CBS scales up to 32 box robots; coupled PRM approaches fail at lower agent counts.
High-DOF Manipulators: CBS-based methods solve up to 8 seven-DOF arms in confined workspaces at an order-of-magnitude speed and success-rate advantage over coupled or decoupled PRM approaches, with lower path cost and fewer roadmap nodes.
Heterogeneous Fleets: CBS generalizes directly to mixed teams (e.g., ground, aerial, and articulated robots), with 100% success in empirical trials versus low success for decoupled methods.
Real-World Deployment: CBS variants have been demonstrated on actual robotic fleets navigating semantic topometric maps in industrial and laboratory environments, including non-holonomic systems (Fredriksson et al., 29 Jan 2025).
Explainable and Safety-Critical Planning: Extensions produce plans with bounded index (segmentation length), facilitating human verification in safety-critical scenarios (Kottinger et al., 2022).

7. Open Problems and Limitations

CBS remains an active area of research with several substantial open problems and inherent limitations:

Infinite State Spaces: In continuous time and space, where waiting durations or motion primitives are not finitely parameterized, the constraint tree may in principle be infinite. Completeness and termination are only guaranteed under suitable discretization or roadmap finiteness (Li et al., 13 Jan 2025, Walker et al., 16 Aug 2024).
Efficiency in Dense Regimes: While many enhancements (meta-agent formation, macro-splitting, heuristics) have improved scalability, exponentially-large constraint trees persist at high agent/space density, motivating ongoing development of more informed search and pruning strategies (Gordon et al., 2021).
Path Quality in High-Dimension: Strategies like artificial potential field post-processing may reduce search work but sometimes increase overall path cost by up to 30% in manipulator settings (Hong et al., 17 Sep 2025).
Multi-Objective and Pareto Front Discovery: Generalization to truly multi-objective planning (with Pareto-front enumeration) is algorithmically tractable but incurs exponential blowup in the number of solution vectors, demanding additional efficiency innovations (Ren et al., 2021).

References:

(Solis et al., 2019) "Representation-Optimal Multi-Robot Motion Planning using Conflict-Based Search" (Tolpin, 2014) "Justifying and Improving Meta-Agent Conflict-Based Search" (Kottinger et al., 2022) "Conflict-based Search for Multi-Robot Motion Planning with Kinodynamic Constraints" (Andreychuk et al., 2021) "Improving Continuous-time Conflict Based Search" (Gordon et al., 2021) "Revisiting the Complexity Analysis of Conflict-Based Search: New Computational Techniques and Improved Bounds" (Fredriksson et al., 29 Jan 2025) "Multi-Agent Path Finding Using Conflict-Based Search and Structural-Semantic Topometric Maps" (Lim et al., 2022) "CBS-Budget (CBSB): A Complete and Bounded Suboptimal Search for Multi-Agent Path Finding" (Hong et al., 17 Sep 2025) "Repulsive Trajectory Modification and Conflict Resolution for Efficient Multi-Manipulator Motion Planning" (Veerapaneni et al., 1 Oct 2025) "Conflict-Based Search as a Protocol: A Multi-Agent Motion Planning Protocol for Heterogeneous Agents, Solvers, and Independent Tasks" (Huang et al., 2020) "Learning to Resolve Conflicts for Multi-Agent Path Finding with Conflict-Based Search" (Yang, 2020) "Resolving Head-On Conflicts for Multi-Agent Path Finding with Conflict-Based Search" (Kottinger et al., 2022) "Conflict-Based Search for Explainable Multi-Agent Path Finding" (Li et al., 13 Jan 2025) "CBS with Continuous-Time Revisit" (Ren et al., 2021) "A Conflict-Based Search Framework for Multi-Objective Multi-Agent Path Finding" (Walker et al., 16 Aug 2024) "On the Completeness of Conflict-Based Search: Temporally-Relative Duplicate Pruning"